In a medium bowl, mix all the dry ingredients: flour, baking powder, salt, sugar, grated orange, and nutmeg.
Stir in the milk, eggs, vanilla, and oil, and mix well. The batter should be smooth, thick, and spoonable.
Pour vegetable oil into a large saucepan at least 3 inches deep (about 5 centimeters), and place on medium heat until the oil is 375℉ (190 ℃).
Scoop up the batter with your fingers or two spoons, one to scoop the dough in the shape of a ball and the other to gently push it into the hot oil. Fry them in batches so you don't overcrowd the pan.
Fry for a few minutes until the bottom side is golden brown. Turn the ball over and fry for a few more minutes until the other side is golden brown.
Remove the doughnuts from the oil with a large slotted spoon or something similar. Place them immediately on paper towels to soak up excess oil. Roll them in table sugar or powdered sugar to make them sweeter if desired.